Ebook: Hybrid switching diffusions: Properties and applications

This book presents a comprehensive study of hybrid switching diffusion processes and their applications. The motivations for studying such processes originate from emerging and existing applications in wireless communications, signal processing, queueing networks, production planning, biological systems, ecosystems, financial engineering, and modeling, analysis, and control and optimization of large-scale systems, under the influence of random environment. One of the distinct features of the processes under consideration is the coexistence of continuous dynamics and discrete events. This book is written for applied mathematicians, applied probabilists, systems engineers, control scientists, operations researchers, and financial analysts. Selected materials from the book may also be used in a graduate level course on stochastic processes and applications or a course on hybrid systems. A large part of the book is concerned with the discrete event process depending on the continuous dynamics. In addition to the existence and uniqueness of solutions of switching diffusion equations, regularity, Feller and strong Feller properties, continuous and smooth dependence on initial data, recurrence, ergodicity, invariant measures, and stability are dealt with. Numerical methods for solutions of switching diffusions are developed; algorithms for approximation to invariant measures are investigated. Two-time-scale models are also examined. The results presented in the book are useful to researchers and practitioners who need to use stochastic models to deal with hybrid stochastic systems, and to treat real-world problems when continuous dynamics and discrete events are intertwined, in which the traditional approach using stochastic differential equations alone is no longer adequate.

This book focuses on switching diffusion processes involving both continuous dynamics and discrete events.

The first part, including three chapters, presents basic properties such as Feller and strong Feller, recurrence, and ergodicity. With a brief review of existence and uniqueness of solutions of switching diffusions, basic properties such as recurrence, Feller properties etc. are dealt with.

The second part of the book is devoted to numerical solutions of switching diffusions.

Containing three chapters, the third part focuses on stability. Chapter seven and chapter eight proceed with the stability analysis. The approach is based on Liapunov function methods.

For convenient references, an appendix including a number of mathematical preliminaries are placed at the end of the book. Topics discussed here including Markov chains, martingales, Gaussian processes, diffusions, jump diffusions, and weak convergence methods. Although detailed developments are often omitted, appropriate references are provided for the reader for further reading.